Study on the distances and arrangement of finite-point-set

1.设G是从具有可数个顶点的完全图中去掉一条边而得到的图。则G不是任意维的整数距离图，而是平面上的有理距离图。如果一个具有n个顶点的完全图可以表示为平面上的一个整数距离图，使得一条直线上没有三个顶点，一个圆上也没有四个顶点，则完全n部图K(a_1，a_2，[三重键]，a_n)是平面上的有理距离图，其中a_k=(k-1选择2)+(k-1选择3)+1.2。1，存在一个n点集，使得p-范数(p=1，2，[三重键]，k)的n点集的极小星的中心都是不同的。但对于任何4点集，极小星的中心对于任何范数都是相同的点。(与M.Watanabe合作)3.设X是至少有m+2个点的点集。从X的m+1点集族到分配给每个(m+1)点集的非负实数，即(m+1)点集的凸包的m维体积的映射是半度量的，并且满足m维单纯形不等式。对于每个m，我们可以定义m维单纯形不等式的“界”S(M)。S(M)的这个界在某种程度上决定了“配置”X。例如，如果|X|&gt；4，则S(2)=2、S(3)=3和[X是正则单纯形的顶点集]三个语句是等价的。计算了三维空间中正多面体的S(M)。虽然对于n维正多面体n&gt；m-1&gt；1，S(M)=3，但当n趋于无穷大时，n-立方体的S(M)的值趋于1。(与M.Deza和M.duour共同工作)。

1.Let G be the graph obtained from a complete graph with countably many vertices by removing an edge. Then G is not an integral-distance graph in any dimension, but it is a rational distance graph in the plane. If a complete graph with n vertices can be realized as an integral distance graph in the plane in such a way that no three vertices lie on a line, and no four vertices lie on a circle, then the complete n-partite graph K(a_1,a_2,【triple bond】,a_n) is a rational distance graph in the plane, where a_k=(k-1 choose 2)+(k-1 choose 3)+1.2.For any n>4, and any k>1, there is an n-point-set such that the center of the minimal star of the n-point-set in p-norm (p=1,2,【triple bond】,k) are all distinct. But for any 4-point-set the center of the minimal star are the same point for any norm. (Joint work with M.Watanabe).3.Let X be a point-set with at least m+2 points. The map from the family of m+1 point-set of X to the nonnegative reals that assigns to each (m+1)-point-set, the m-dimensional volume of the convex hull of the (m+1)-point-set, is a hemimetric and satisfies the m-dimensional simplex inequality. For each m, we can define the "bound" s(m) of m-dimensional simplex inequality. This bound s(m) determines the "configuration" X to some extent. For example, if |X|>4, then the three statements s(2)=2,s(3)=3, and [X is the vertex-set of a regular simplex] are equivalent. We calculated s(m) for regular polyhedra in 3-space. Though s(m)=3 for the n-dimensional cross-polytope, n>m-1>1, the value s(m) for n-cube tends to 1 as n tends to infinity. (Joint work with M.Deza and M.Dutour).